0132_0321_1203
Counting sequence:
1, 1, 2, 6, 21, 74, 257, 886, 3050, 10505, 36206, 124833, 430474, 1484526, 5119597, 17655746, 60888801, 209985534, 724171922, 2497434881, 8612847430, 29702935745, 102435859346, 353268294998, 1218308608389, 4201554149626, 14489807575617, 49970681390086, 172332792312506, 594320319146169, 2049619442941662, 7068477596533601, 24376903578691610, 84068092453346110, 289923785683328861, 999853797702756338, 3448173851708055329, 11891641496912662926, 41010443084554110562, 141431815147491748401, 487752797370990456886, 1682102369223887609729, 5801029529301731725602, 20005883242027425659046, 68993850534977490750965, 237937578364094113351850, 820570105288846952688001, 2829882115818654235171478, 9759352354923172187430794, 33656864310756328395005801, 116071689394555179472783182, 400293888180201057039025441, 1380484748264031038054097066, 4760847458484540751143075214, 16418630159776081079568311117, 56622569547590154997222514466, 195273013093767701557127867745, 673433755256725949698622562590, 2322456214169258936650297161266, 8009403782673746346285923421793, 27621855069872706231206090090534, 95258885455560641980649975464001, 328517227944366338083626167271602, 1132950155149551154190675988937206, 3907180339019428173464581801944421, 13474607097436542238773628200662874, 46469581815067691609387715055947457, 160258627094075798114673296316872486, 552680410602499278418639617862095450, 1906016804227563641410435219904871705, 6573238327802288696450476230195529406, 22668982780348243078098027945513848033, 78178023474700009388058612322706639674, 269610834047175345848608942579782274526, 929800967136723281559074433465466395197, 3206584192151248082157400393023767620946, 11058476539358472275228579780364948087201, 38137125378111252766909185533776408092334, 131522667424327944716437380036493126405122, 453579337060854056467531918470973052334481, 1564249106542290887187505275794895438423830, 5394591568420746925074621867447107908146945, 18604209565063558963769933414236582760532546, 64159929283048055368475775126919500744272198, 221266940216369458786971825845755589589829589, 763078441323190168230671741826128715752975626, 2631611875876391399563291480088397256540105217, 9075582129204094155657803464501673323488463286, 31298760937723293364328685752093605196235524906, 107939350037336177981744633813371179306830016969, 372248067892047525040053421412743184649226050862, 1283763743263523376112039973171155683145550871201, 4427287850949733448477561518866665603045663832010, 15268290460781114119363564390678792502419106827310, 52655418270300846632194683611555501583079894742061, 181591585537500211532247499739275283849992102097538, 626250916263680655210032546531580527156138559298529, 2159737792696947049359447239256829262199850218084926, 7448240333175854606368527726303166752646726905694162, 25686582995555312950669481903202622272446338340209601, 88584755119766530387360606602360052349521918094593286
Generating function in Maple syntax:
(3*x-1)*(-1+x)^3/(9*x^4-19*x^3+17*x^2-7*x+1)
Generating function in latex syntax:
\frac{\left(3 x -1\right) \left(-1+x \right)^{3}}{9 x^{4}-19 x^{3}+17 x^{2}-7 x +1}
Generating function in sympy syntax:
(x - 1)**3*(3*x - 1)/(9*x**4 - 19*x**3 + 17*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(9*x^4-19*x^3+17*x^2-7*x+1)*F(x)-(3*x-1)*(-1+x)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(9 x^{4}-19 x^{3}+17 x^{2}-7 x +1\right) F \! \left(x \right)-\left(3 x -1\right) \left(-1+x \right)^{3} = 0
Explicit closed form in Maple syntax:
-288/491*Sum(_alpha^(-n+2),_alpha = RootOf(9*_Z^4-19*_Z^3+17*_Z^2-7*_Z+1))+341/491*Sum(_alpha^(-n+1),_alpha = RootOf(9*_Z^4-19*_Z^3+17*_Z^2-7*_Z+1))-100/491*Sum(_alpha^(-n),_alpha = RootOf(9*_Z^4-19*_Z^3+17*_Z^2-7*_Z+1))+29/491*Sum(_alpha^(-n-1),_alpha = RootOf(9*_Z^4-19*_Z^3+17*_Z^2-7*_Z+1))+piecewise(n = 0,1/3,0)
Explicit closed form in latex syntax:
-\frac{288 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(9 \textit{\_Z}^{4}-19 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{491}+\frac{341 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(9 \textit{\_Z}^{4}-19 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{491}-\frac{100 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(9 \textit{\_Z}^{4}-19 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{491}+\frac{29 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(9 \textit{\_Z}^{4}-19 \textit{\_Z}^{3}+17 \textit{\_Z}^{2}-7 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{491}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+4) = -9*a(n)+19*a(n+1)-17*a(n+2)+7*a(n+3), n >= 5
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 21
a \! \left(n +4\right) = -9 a \! \left(n \right)+19 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14699/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[12,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[58,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[6,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[20,x]+F[31,x]
F[31,x] = 2*F[15,x]+F[32,x]+F[36,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[24,x]
F[35,x] = F[31,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]+F[53,x]
F[39,x] = F[15,x]+F[40,x]+F[51,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[10,x]+F[43,x]
F[43,x] = F[29,x]
F[44,x] = F[39,x]+F[45,x]
F[45,x] = 2*F[15,x]+F[46,x]+F[50,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[43,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[38,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[2,x]+F[39,x]
F[53,x] = 2*F[15,x]+F[50,x]+F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[28,x]
F[57,x] = F[53,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[52,x]+F[60,x]
F[60,x] = F[39,x]+F[61,x]
F[61,x] = F[15,x]+F[54,x]+F[62,x]+F[64,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[39,x]
F[64,x] = F[12,x]*F[60,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{58}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{24}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{53}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{51}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{29}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{28}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{39}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{39}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_12(x)*F_4(x))
Eq(F_4(x), F_19(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_58(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_6(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_20(x) + F_31(x))
Eq(F_31(x), 2*F_15(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_24(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x) + F_53(x))
Eq(F_39(x), F_15(x) + F_40(x) + F_51(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_10(x) + F_43(x))
Eq(F_43(x), F_29(x))
Eq(F_44(x), F_39(x) + F_45(x))
Eq(F_45(x), 2*F_15(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_43(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_38(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_2(x) + F_39(x))
Eq(F_53(x), 2*F_15(x) + F_50(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_28(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_52(x) + F_60(x))
Eq(F_60(x), F_39(x) + F_61(x))
Eq(F_61(x), F_15(x) + F_54(x) + F_62(x) + F_64(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_39(x))
Eq(F_64(x), F_12(x)*F_60(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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